Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{a^2 + 9a}{a^2 - 81}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{a^2 + 9a}{a^2 - 81} = \dfrac{(a)(a + 9)}{(a - 9)(a + 9)} $ Notice that the term $(a + 9)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a + 9)$ gives: $z = \dfrac{a}{a - 9}$ Since we divided by $(a + 9)$, $a \neq -9$. $z = \dfrac{a}{a - 9}; \space a \neq -9$